Geometric constructions for Ramsey-Tur\'an theory

TL;DR

This paper investigates the structure of extremal graphs in Ramsey-Turán theory, refutes a longstanding conjecture about their periodicity, and introduces new constructions using high-dimensional spheres with rational densities.

Contribution

It disproves the conjecture that extremal structures are periodic and constructs Bollobás-Erdős-type graphs with rational densities using high-dimensional complex spheres.

Findings

Asymptotic extremal structures differ significantly from the p=2 case.

Constructed Bollobás-Erdős-type graphs with various rational densities.

Provided matching upper bounds for the constructed graphs.

Abstract

Combining two classical notions in extremal combinatorics, the study of Ramsey-Tur\'an theory seeks to determine, for integers $m\le n$ and $p \leq q$, the number $\mathsf{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex $K_q$-free graph in which every set of at least $m$ vertices contains a $K_p$. Two major open problems in this area from the 80s ask: (1) whether the asymptotic extremal structure for the general case exhibits certain periodic behaviour, resembling that of the special case when $p=2$; (2) constructing analogues of Bollob\'as-Erd\H{o}s graphs with densities other than $1/2$. We refute the first conjecture by witnessing asymptotic extremal structures that are drastically different from the $p=2$ case, and address the second problem by constructing Bollob\'as-Erd\H{o}s-type graphs using high dimensional complex spheres with all rational densities. Some matching upper bounds are also provided.